Listing Small Minimal Separators of a Graph

TL;DR

This paper introduces an algorithm for enumerating small minimal separators in a graph, achieving fixed-parameter-delay and incremental-polynomial time, filling a gap in existing graph enumeration methods.

Contribution

The paper presents the first non-trivial time complexity algorithm for enumerating minimal $a,b$-separators of size at most $k$, with fixed-parameter-delay and incremental-polynomial time.

Findings

Algorithm enumerates minimal separators efficiently

Achieves fixed-parameter-delay and incremental-polynomial time

No prior algorithms with non-trivial complexity for this problem

Abstract

Let $G$ be a graph and $a,b$ vertices of $G$. A minimal $a,b$-separator of $G$ is an inclusion-wise minimal vertex set of $G$ that separates $a$ and $b$. We consider the problem of enumerating the minimal $a,b$-separators of $G$ that contain at most $k$ vertices, given some integer $k$. We give an algorithm which enumerates such minimal separators, outputting the first $R$ minimal separators in at most $poly(n) R \cdot \min(4^k, R)$ time for all $R$. Therefore, our algorithm can be classified as fixed-parameter-delay and incremental-polynomial time. To the best of our knowledge, no algorithms with non-trivial time complexity have been published for this problem before. We also discuss barriers for obtaining a polynomial-delay algorithm.